Integrand size = 35, antiderivative size = 694 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\frac {\left (\frac {b^2 e^2}{a}+a f^2+b \left (\frac {d e^2}{c}+12 e f+\frac {c f^2}{d}\right )\right ) x \sqrt {c+d x^2}}{3 \sqrt {a+b x^2}}+\frac {(b e (d e+6 c f)+a f (6 d e+13 c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c}+\frac {2 f (3 b e (d e+2 c f)+a f (6 d e+5 c f)) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e}+\frac {f^2 (a f (10 d e+3 c f)+2 b e (6 d e+5 c f)) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e^2}+\frac {f^3 (10 b d e+3 b c f+3 a d f) x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e^2}+\frac {b d f^4 x^9 \sqrt {a+b x^2} \sqrt {c+d x^2}}{a c e^2}-\frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3}{3 a c e x^3}-\frac {f \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3}{a c e^2 x}-\frac {\left (b^2 c d e^2+a^2 c d f^2+a b \left (d^2 e^2+12 c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} \sqrt {b} c d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {2 \sqrt {a} (a f (3 d e+c f)+b e (d e+3 c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*(b^2*e^2/a+a*f^2+b*(d*e^2/c+12*e*f+c*f^2/d))*x*(d*x^2+c)^(1/2)/(b*x^2+ a)^(1/2)+1/3*(b*e*(6*c*f+d*e)+a*f*(13*c*f+6*d*e))*x*(b*x^2+a)^(1/2)*(d*x^2 +c)^(1/2)/a/c+2/3*f*(3*b*e*(2*c*f+d*e)+a*f*(5*c*f+6*d*e))*x^3*(b*x^2+a)^(1 /2)*(d*x^2+c)^(1/2)/a/c/e+1/3*f^2*(a*f*(3*c*f+10*d*e)+2*b*e*(5*c*f+6*d*e)) *x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2+1/3*f^3*(3*a*d*f+3*b*c*f+10*b *d*e)*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2+b*d*f^4*x^9*(b*x^2+a)^(1 /2)*(d*x^2+c)^(1/2)/a/c/e^2-1/3*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^ 3/a/c/e/x^3-f*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^3/a/c/e^2/x-1/3*(b ^2*c*d*e^2+a^2*c*d*f^2+a*b*(c^2*f^2+12*c*d*e*f+d^2*e^2))*(d*x^2+c)^(1/2)*E llipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/b^ (1/2)/c/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+2/3*a^(1/2)*(a*f *(c*f+3*d*e)+b*e*(3*c*f+d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/ 2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/ (b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 2.40 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (c+d x^2\right ) \left (b c e^2 x^2+a d e^2 x^2+a c \left (e^2+6 e f x^2-f^2 x^4\right )\right )-i c \left (b^2 c d e^2+a^2 c d f^2+a b \left (d^2 e^2+12 c d e f+c^2 f^2\right )\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (b d e^2+a f (6 d e+c f)\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^4,x]
Output:
(Sqrt[b/a]*(-(Sqrt[b/a]*d*(a + b*x^2)*(c + d*x^2)*(b*c*e^2*x^2 + a*d*e^2*x ^2 + a*c*(e^2 + 6*e*f*x^2 - f^2*x^4))) - I*c*(b^2*c*d*e^2 + a^2*c*d*f^2 + a*b*(d^2*e^2 + 12*c*d*e*f + c^2*f^2))*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d* x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d )*(b*d*e^2 + a*f*(6*d*e + c*f))*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c ]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*c*d*x^3*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])
Time = 1.12 (sec) , antiderivative size = 639, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {448, 442, 403, 27, 406, 320, 388, 313, 442, 406, 320, 388, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^2}dx}{e^2}+e \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^4}dx\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {f \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (3 b e+a f) x^2+2 b c e+a d e+a c f\right )}{\sqrt {d x^2+c}}dx}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {a d \left ((6 b d e+b c f+a d f) x^2+3 b c e+3 a d e+2 a c f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (\frac {\frac {1}{3} a \int \frac {(6 b d e+b c f+a d f) x^2+3 b c e+3 a d e+2 a c f}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (\frac {\frac {1}{3} a \left ((2 a c f+3 a d e+3 b c e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(a d f+b c f+6 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (\frac {\frac {1}{3} a \left ((a d f+b c f+6 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (\frac {\frac {1}{3} a \left ((a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}+e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle e \left (\frac {\frac {\int \frac {b d (b c e+a d e+6 a c f) x^2+a c (2 b d e+3 b c f+3 a d f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle e \left (\frac {\frac {a c (3 a d f+3 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b d (6 a c f+a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle e \left (\frac {\frac {b d (6 a c f+a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle e \left (\frac {\frac {b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )+\frac {f \left (\frac {\frac {1}{3} a \left (\frac {\sqrt {c} \sqrt {a+b x^2} (2 a c f+3 a d e+3 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f+b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (a f+3 b e)}{a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{a x}\right )}{e^2}\) |
Input:
Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^4,x]
Output:
(f*(-((e*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x)) + (((3*b*e + a*f)*x*Sqr t[a + b*x^2]*Sqrt[c + d*x^2])/3 + (a*((6*b*d*e + b*c*f + a*d*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan [(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/( a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(3*b*c*e + 3*a*d*e + 2*a*c*f) *Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/ (a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/3)/a)) /e^2 + e*(-1/3*(e*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^3) + (-((a*(d*e + 3*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x)) + (b*d*(b*c*e + a*d*e + 6 *a*c*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2 ]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt [(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(2*b*d*e + 3*b*c*f + 3*a*d*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/c)/(3*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e Int[(g*x)^m*(a + b*x ^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
Time = 5.92 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.64
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 x^{3}}-\frac {e \left (6 a c f +a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a c x}+\frac {f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3}+\frac {\left (\frac {2}{3} a c \,f^{2}+2 a d e f +2 b c e f +\frac {2}{3} b d \,e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f +\frac {b d e \left (6 a c f +a d e +b c e \right )}{3 a c}-\frac {f^{2} \left (2 a d +2 b c \right )}{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(444\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-a c \,f^{2} x^{4}+6 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}\right )}{3 x^{3} a c}+\frac {\left (-\frac {\left (a^{2} c d \,f^{2}+a b \,c^{2} f^{2}+12 a b c d e f +a b \,d^{2} e^{2}+b^{2} d \,e^{2} c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {2 a^{2} c^{2} f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {2 a b c d \,e^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {6 a b \,c^{2} e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {6 a^{2} c d e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a c \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(665\) |
| default | \(\text {Expression too large to display}\) | \(1090\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^4,x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/3*e^2*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/3*e*(6*a*c*f+a*d*e+b*c*e)/a/c*(b*d*x ^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+1/3*f^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 /2)+(2/3*a*c*f^2+2*a*d*e*f+2*b*c*e*f+2/3*b*d*e^2)/(-b/a)^(1/2)*(1+b*x^2/a) ^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*( -b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(a*d*f^2+b*c*f^2+2*d*b*e*f+1/3*b*d*e *(6*a*c*f+a*d*e+b*c*e)/a/c-1/3*f^2*(2*a*d+2*b*c))*c/(-b/a)^(1/2)*(1+b*x^2/ a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(Elliptic F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a *d+b*c)/c/b)^(1/2))))
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^4,x, algorithm="fr icas")
Output:
integral((f^2*x^4 + 2*e*f*x^2 + e^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/x^4, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}{x^{4}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2/x**4,x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2/x**4, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^4,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^4, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^4,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^4, x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2}{x^4} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^4,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^4, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^4} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^4,x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2 + 2*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a**2*d**2*f**2*x**2 - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a* b*c**2*f**2 - 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e*f + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**2 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*e**2 + 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*e*f *x**2 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f**2*x**4 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x**2 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e**2 + 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e *f*x**2 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f**2*x**4 - 3*int(( sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c **2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8 ),x)*a**4*c**2*d**2*f**2*x**3 - 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/ (a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d** 2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a**3*b*c**3*d*f**2*x**3 - 36*i nt((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a *b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d* x**8),x)*a**3*b*c**2*d**2*e*f*x**3 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x* *2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a* b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a**3*b*c*d**3*e**2*x**3 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x...